Problem: When an integer is divided by 15, the remainder is 7. Find the sum of the remainders when the same integer is divided by 3 and by 5.
Solution: We let our integer be $n$.  the first sentence tells us that  \[n\equiv 7\pmod {15}.\] Since 3 and 5 are both factors of 15 we deduce \begin{align*}
n&\equiv7\equiv1\pmod3\\
n&\equiv7\equiv2\pmod5.
\end{align*} Therefore the remainders in question are 1 and 2, and their sum is $\boxed{3}$.